I'm having a thought that I would like more expert opinion on, as this crosses boundaries between philosophical logic and political science.

My premise is that a political ideology can be represented as (or, I prefer, actually is) a logical system. We know that certain logical systems are undecidable such as first-order predicate logic. Logical systems can have other meta-logical properties as well, such as soundness, consistency, completeness, and their respective negations. So this challenges my understanding of meta-logic as well, which I'm not completely fluent in.

Examples of political ideologies which I would represent as logical systems are things like Marxism, as well as particular systemizations of feminism, scientific racism, etc (Note: The intent isn't to equivocate any of these, just that these are the best examples I can think of that are clear enough to understand formally. I wouldn't consider things like liberalism or conservatism as ideologies, because they aren't based on a system of ideas, but seem to represent a coalition of interests. Valid?). Scientific theories can also be formalized, but differ from ideologies they don't aim to change things politically or culturally (I know that in some quarters this is controversial).

If someone wants a reference, I'll dig for it, but I've recently read that basically the more expressive a logical system is, the more difficult are it's meta-logical properties...I think decidability itself is mentioned here. Another premise to my thinking is that when you analyze a body of work (an ideology in this case) into a logical system, the more expressive that logical system is the more arguments you might find that are valid. For instance, "All men are mortal. Socrates is a man. Therefore, Socrates is mortal." is invalid if analyzed at the level of propositional logic. I assume that this pattern continues the more expressive the logic is, as long as the more expressive logic is a strict superset of it. For instance, second-order logic will find more validities than first-order logic, etc. I beg for some charity in trying to understand my gist here.

So my question is whether this argument makes sense either logically or philosophically. Basically, I see most of the political ideologies that we know of are either logically undecidable, or are on the path to becoming undecidable, as the ideology is refined through iteration, dealing with contradictions (either external or internal) and political opposition by becoming more subtle and nuanced in the ideas and propositions they express. For instance, Marxism itself represents an advanced iteration of intellectualized socialism. I can go to white supremacist websites and find someone with lengthy logical attempts to argue for the truth of their views.

My issue is with the idea I said before that an argument's validity depends on the logical system, and you can often analyze the same argument using more or less expressive logical systems. I'll use the term "correct" to mean that the argument was found valid after the "best" analysis of the natural language argument. So what does it mean for an argument to be "correct" when it requires an undecidable logical system to analyze the statement? Well, it means that we may never be able to prove that the argument is invalid, even if the proponents of the ideology assert the truth and rationality of it in the best conscience?

I also wonder about arguments that require something like second-order logic, or something of similar complexity, that we may say are sound, but are incomplete. So the ideology might be right, but we can never prove it? Or what if the ideology is logically inconsistent, would we ever be able to establish that?

It seems to me, that if there is any basis to this, then it justifies quite a bit of skepticism of a political process based on rational argument and logical debate. Am I right?

Sorry for making this post so long. I didn't have enough time to make it shorter.
– Kevin HolmesAug 24 '14 at 4:28

Possibly Marxism as presented by Marx has to be diffentiated from Marxism as a political ideology, which as most political ideologies represents a coalition of interests centred on Marx & others such as Rosa Luxembourg, Lenin and so on.
– Mozibur UllahAug 24 '14 at 5:38

I was trying to say by Marxism similar as if you say Peano arithmetic -- not just any formal arithmetic with roughly the same aims, but that system that Peano published at that stage of refinement, only not subject to metalogical constraints.
– Kevin HolmesAug 24 '14 at 6:00

I'm going to pour cold water on your question. Concepts of logic should stay in logic, where they are defined in careful and precise ways. Let's not underestimate just how much machinery needs to be set up to get anything going in logic. Once you step away from that machinery, things get fuzzy and confusing and confused. I doubt any logician would care whether some powerful result (Gödel) applies to Marxism or some other political theory. Nor should they.
– user10450Oct 17 '14 at 4:40

3 Answers
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I think there's a much deeper and more fiddly issue going on here: political ideology requires actual people to implement. There might be issues with logical inconsistency in ideology, but most times you never find it because of the much greater disparity between premises for what people are like and what people empirically are actually like.

Additionally, political ideologies typically don't frame themselves in an idealized form requiring the selection of e.g. a particular type of predicate logic.

So although formally we might entertain the idea of a political ideology being "right", practically we don't care. We don't have the ideology in the form it needs to be to make that judgment, and if it was right in some sense (about some model) the rightness wouldn't translate to the real world because the real world isn't the model (we don't know enough about important aspects like how rational and irrational responses to various social situations are triggered).

Instead, it's more productive to ask whether the ideology is a good match for actual people. If it has the word "ideology" attached, chances are good that the answer is "no", and the follow-up is to figure out just how bad and in which ways.

As I'm trying to figure out why I'm not seeming to get much traction here, the first part of your comment reminds me of the distinction Peirce makes between logica utens and logica docens. Maybe part of the divide is that I'm equivocating the two, or maybe the "deeper issue" are differing intuitions of how they relate, e.g. black2.fri.uni-lj.si/humbug/files/doktorat-vaupotic/zotero/…
– Kevin HolmesAug 24 '14 at 20:18

@KevinHolmes - I'm not sure the distinction is clear to begin with. If logica utens assumes you already know things, and you don't know things (and with ideology's impact on real people, you mostly don't), then you can't even get off the ground. I'm not sure you're equivocating the two so much as being in danger of not fulfilling the prerequisites. (Also, you can't get anywhere with scientific study without a boatload of logical reasoning that connects your observations to some explanation.)
– Rex KerrAug 24 '14 at 23:15

If they are the same, then everyone (people in ideologies) wants to be logically consistent at some level, and to give only valid arguments. This is the basis of believing that ideologies can implicitly represent logical systems. Even if the axioms, what constitutes a valid inference, differ, this can be modelled as simply a variation of a calculus. But while ideologies may be consistent at the local level (logica utens), they don't need to be at the system level (logica docens). This makes modeling ideologies as logical systems problematic.
– Kevin HolmesAug 25 '14 at 0:31

Btw, I can actually leave the issue of ideologies being "right" aside. My motivation springs from the current political climate of at least rhetorical turmoil and polarization. The democratic ideal of opposing views entering into rational discourse to establish consensus seems as far away as ever, and I'm trying to explain this.
– Kevin HolmesAug 25 '14 at 0:38

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@KevinHolmes - If you want to explain that I think your first stop should be the study of in-group out-group interactions, not logic.
– Rex KerrAug 25 '14 at 19:46

A well-known example of axiomatized system is Spinoza's Ethics; but it is not formalized.

In order to formalize a "system" we have to :

choose a language, with an alphabet, i.e. a finite set of symbols,

define a grammar, i.e. a set of rules defining what is a "permissible" statement of the system

choose a set of axioms

choose a set of logical axioms and inference rules; this is the "easy task" : except fro Hegel, we can use classical logic.

In our system we have some primitive terms (like : set and membership in set-theory) which are undefined (it is impossible to define all ...) and which "receive meaneing" through the axioms.

Then we have defined terms (like : ordered pair and relation in set-theory) which are introduced through explicit definition satisfying some precise rules.

Only after having done this we can speak about "logical" properties of a system/theory, like : consistency, soundness, completeness and decidability, where :

a theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory.

In the context of a formalized political ideology, to ask for decidability means to ask for a method for "finding an answer" to every problem expressible in the system.

The last clause is the key fact : we must be able to express the problem in the language of the system/theory, i.e. through a "permissible" statement according to the grammar of the system.

Only if we can reach this point, we can ask if there is a "decision method" for the formalized system/theory. This is not a trivial issue : in mathematics, only some formalized theories are decidable.

Having said this, what about the question :

So the ideology might be right, but we can never prove it?

The issue is : how to "formalize" the statement that "the ideology is right" ?

Possibly Marxism as presented by Marx has to be diffentiated from Marxism as a political ideology, which as most political ideologies represents a coalition of interests centred on Marx & others such as Rosa Luxembourg, Lenin and so on; one might also want to reach for Simone Weil who recognised that force as a deciding factor in politics in an essay on the Illiad.

I find it a little suprising that something so human as politics can be decided by formal logic; classically politics is distinguished by Rhetoric as persuasive speech as taught by the Sophists, and violence - or its negation - as in the non-violent ahimsa of Gandhi.

Proof isn't the measure of an ideology but how persuasive it is; a skillful orator intertwines appeals to emotions through symbols and reason through argument.

I suppose I was latching onto the possibility that politics could be rational, and that there could be a correct view beyond my own self-interest. Otherwise, it seems to me about manipulating the public.
– Kevin HolmesAug 24 '14 at 15:20

@holmes: I agree that this is important; but abstracting a reasoned argument into formal logic, I think, changes what is being argued; for xample to take Hegels dialectic and place it into formal logic is to take p as the thesis; to take -p as the antithesis; and we can write p+(-p) as the sublated synthesis of the thesis and the synthesis; but what is this 'sum'; formally we could say that it is just zero; this would be the usual mathematical interpretation;
– Mozibur UllahAug 24 '14 at 17:12

but what this misses out on is that the sublation depends intrinsically on the quality of the thesis and anti-thesis which can't be captured by formalising them; for example Hegel (a la the Tao) begins with p as nothing; and -p as Being; and the synthesis as Becoming.
– Mozibur UllahAug 24 '14 at 17:14

This doesn't of course mean that logic & mathematics can't be useful in the social, political & philosophical sphere; for example one might look at trying to see what extent the Labour Theory of Value is true; or rather provide evidence towards it;
– Mozibur UllahAug 24 '14 at 17:17

I don't think it is impossible to logically analyze a system such as Hegel's, other than that the resulting complexity would be undecidable, incomplete, and possibly inconsistent. So you would be right in the sense that such an analysis would lose most of its deductive power, and may not even qualify as logic. Also...your formalization of Hegel isn't correct because the formalism you're trying to use isn't expressive enough.
– Kevin HolmesAug 24 '14 at 17:30